The recent crowd size debate, (if we can call this a debate), on the inauguration reminded me of something that happens at every demonstration in France. The estimations of the number of demonstrators, according to the police, and according to the organisers, are very different, with a ratio as large as 1 to 10.

Here is a 6/14/16 Le Monde article (in French) about this. It concerns a demonstration about the last labor legislation; the police said there were 125,000 demonstrators, while the organizers (unions) said there were 1,300,000 demonstrators, (1.3 million, there is more than a factor 10 between their estimations, I did not type too many zeros).

Question: What are the different techniques employed to estimate the size of a crowd? Who uses them?

I am not looking for a visually vague answer like a "picture from above". If it is with a picture from above, what do we do with this picture? Are the people counted by hand in a small area and then an extrapolation is made? Is this area representative? Are pictures taken at different moments considered? Is the time the crowd needs to pass a given point considered?

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    This is really interesting, but it is not about politics. – Alexei Feb 6 '17 at 14:03
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    I don't have time to hunt for references, but none of the existing answers mentioned alternate methods (internally generated, such as organizers giving out pins/etc... and counting how many were given); and then there's a related question discussed on a resent 538 podcast over the accuracy of internal estimates by organizations. – user4012 Feb 6 '17 at 15:31
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    I think this question is on the verge, but still barely on-topic. Demonstrations are a form of political participation and their effect on politics usually depends on the number of participants. But I don't feel strong enough about this to overrule the consensus of the community. – Philipp Feb 6 '17 at 16:50
  • This is a serious enough policy question; certainly the military and police are interested both in estimating, (and perhaps at budget or election time mis-estimating), crowd sizes. Citizens are also interested in accurate estimates. Vote to reopen. – agc Feb 7 '17 at 5:20

This is hard to generalize, because there are so many different organizations which organize demonstrations and other public gatherings and then report their estimations regarding participations. Also, there are countless police departments in the world, and they all also use different methods.

But a more reliable method to count crowd sizes is the Jacob's Method. In order to use this method, you need:

  • An exact map of the area where the crowd gathered
  • A photo which shows the whole area with the crowd (like an aerial photo or one from a high vantage point) which is large enough to see which parts are covered and which aren't but still detailed enough to estimate how tightly grouped the people are.

The method works as follows

  1. You divide the map of the assembly area into smaller polygons. The vertices of your polygons need to be landmarks which you can identify both on your map and in your photo.
  2. You calculate the area in m² for each polygon
  3. You map those polygons on your photograph as well as possible
  4. For each polygon, you estimate how much of it is filled by people (fill-factor) and estimate the number of people per m² (density). The numbers from Jacob's original paper were:
    • standing tightly: 4.3
    • crowded but not pushing each other: 2.4
    • standing in a distance of one person’s arm: 1.1

Now you can calculate the number of people in each polygon by multiplying the area in m² with the fill-factor and the estimated density. Do that for all polygons, add up the numbers and you should have a quite good estimation.

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The proper estimation of the crowd size is an important factor for police and emergency services. For instance, knowing the number of attendees and exits in a closed location is useful for prediction panic in an emergency case (like fire or shooting). It is also important to know the crowd size if you want to know how many police and guards you need to ensure the security of an event.

One of the most known methods of counting is named after its inventor, Herbert Jacobs.

"I would say the most accurate way to estimate the number of people in a large outdoor crowd is to get good aerial imagery, break it into regions of similar density, measure the square footage of each region, apply a reasonable value for the density in each region, and then add the calculated number of people in each region back into a total estimate," said Stephen Doig, a census expert at Arizona State University.
"The only better way would be if the crowd had to buy tickets and go through turnstiles to get in; alas, that rarely is the case," Doig told Life's Little Mysteries. — (source)

In this article on NBC News, Stephen Doig says:

A loose crowd, one where each person is an arm's length from the body of his or her nearest neighbors, needs 10 square feet per person.
A more tightly packed crowd fills 4.5 square feet per person.
A truly scary mob of mosh-pit density would get about 2.5 square feet per person.

AFAIR (unsourced), the police of Ukraine uses a similar methodology of splitting the crowd into areas of different densities and then calculating the number of areas of each density. It also has internal instructions that refer different crowd density of ½ to 6 people per square meter. This technique was used to calculate the number of people during the Revolution of Dignity of 2014.

Revolution of Dignity, Kyiv, Ukraine, 2014 Revolution of Dignity, Kyiv, Ukraine, 2014

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